Chapter 16
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Finance
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Jan 9, 2024
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1.
Award: 10.00
points
Problems? Adjust credit for all students.
A 9-year bond paying coupons annually has a yield of 10% and a duration of 7.194 years. Assume the market yield changes by 50 basis points.
Required:
What is the percentage change in the bond’s price?
Note: Assume modified duration and a positive increase in yield change. Do not round intermediate calculations. Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.
The percentage change in the bond’s price is
(3.27)
%
Explanation:
Assume the 7.194 is the Macaulay duration (not Modified Duration). The percentage change in the bond’s price is:
− (D ÷ (1 + y
)) × Δ
y
= −(7.194 ÷ 1.10) × 0.005 = −0.0327 = −3.27%, or a 3.27% decline
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
2.
Award: 10.00
points
Problems? Adjust credit for all students.
Required:
a. Find the duration of a 6% coupon bond making annual
coupon payments if it has three years until maturity and has a yield to maturity of 6%.
Note: The face value of the bond is $1,000. Do not round intermediate calculations. Round your answer to 2 decimal places.
b. What is the duration if the yield to maturity is 10%?
Note: The face value of the bond is $1,000. Do not round intermediate calculations. Round your answer to 2 decimal places.
a. 6% YTM
2.83
years
b. 10% YTM
2.82
years
Explanation:
a. YTM = 6%
(1)
Time until Payment
(Years)
(2)
Cash Flow
(3)
PV of CF (Discount
rate = 6%)
(4)
Weight
(5)
Column (1) ×
Column (4)
1
$ 60.00
$ 56.60
0.0566
0.0566
2
60.00
53.40
0.0534
0.1068
3
1,060.00
890.00
0.8900
2.6700
Column Sums:
$ 1,000.00
1.0000
2.8334
Duration = 2.83 years
b. YTM = 10%
(1)
Time until
Payment (Years)
(2)
Cash Flow
(3)
PV of CF
(Discount rate =
10%)
(4)
Weight
(5)
Column (1) ×
Column (4)
1
$ 60.00
$ 54.55
0.0606
0.0606
2
60.00
49.59
0.0551
0.1101
3
1,060.00
796.39
0.8844
2.6531
Column Sums:
$ 900.53
1.0000
2.8238
Duration = 2.82 years, which is less than the duration at the YTM of 6%.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
3.
Award: 10.00
points
Problems? Adjust credit for all students.
Find the duration of a 6% coupon bond making semiannually coupon payments if it has three years until maturity and has a yield to maturity of 6%. What is the duration if the yield to maturity is 10%?
Note: The face value of the bond is $1,000. Do not round intermediate calculations. Round your answers to 2 decimal places.
Duration
6% YTM
2.79
years
10% YTM
2.78
years
Explanation:
For a semiannual 6% coupon bond selling at par, we use the following parameters: coupon = 3% per half-year period, y
= 3%, T
= 6 semiannual periods.
(1)
Time until Payment (Years)
(2)
Cash Flow
(3)
PV of CF (Discount rate = 3%)
(4)
Weight
(5)
Column (1) × Column (4)
1
$ 30.00
$ 29.13
0.02913
0.0291
2
30.00
28.28
0.02828
0.0565
3
30.00
27.45
0.02745
0.0824
4
30.00
26.65
0.02665
0.1066
5
30.00
25.88
0.02588
0.1294
6
1,030.00
862.61
0.86261
5.1757
Column Sums:
$ 1,000.00
1.00000
5.5797
D
= 5.5797 half-year periods = 2.79 years
If the bond’s yield is 10%, use a semiannual yield of 5% and semiannual coupon of 3%:
(1)
Time until Payment
(Years)
(2)
Cash Flow
(3)
PV of CF (Discount rate =
5%)
(4)
Weight
(5)
Column (1) × Column
(4)
1
$ 30.00
$ 28.57
0.03180
0.0318
2
30.00
27.21
0.03029
0.0606
3
30.00
25.92
0.02884
0.0865
4
30.00
24.68
0.02747
0.1099
5
30.00
23.51
0.02616
0.1308
6
1,030.00
768.60
0.85544
5.1326
Column Sums:
$ 898.49
1.00000
5.5522
D
= 5.5522 half-year periods = 2.78 years
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
4.
Award: 10.00
points
Problems? Adjust credit for all students.
You predict that interest rates are about to fall. Which bond will give you the highest capital gain?
Zero coupon, long maturity
Explanation:
Investors tend to purchase longer term bonds when they expect yields to fall so that they can capture significant capital gains, and the lack of a coupon payment ensures the capital gain will be even greater.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
5.
Award: 10.00
points
Problems? Adjust credit for all students.
In each of the following pairs of bonds, select the bond that has the highest duration or effective duration:
Required
a. Bond A
is a 6% coupon bond, with a 20-year time to maturity selling at par value. Bond B
is a 6% coupon bond, with a 20-year time to maturity selling below par value.
b. Bond A
is a 20-year noncallable coupon bond with a coupon rate of 6%, selling at par. Bond B
is a 20-year callable bond with a coupon rate of 7%, also selling at par.
a. Highest duration
Bond A
b. Highest duration
Bond A
Explanation:
a. Bond B has a higher yield to maturity than bond A since its coupon payments and maturity are equal to those of A, while its price is lower. (Perhaps the yield is higher because of differences in credit risk.) Therefore, the
duration of Bond B must be shorter.
b. Bond A has a lower yield and a lower coupon, both of which cause Bond A to have a longer duration than Bond B. Moreover, A cannot be called, so that its maturity is at least as long as that of B, which generally increases
duration.
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
6.
Award: 10.00
points
Problems? Adjust credit for all students.
An insurance company must make payments to a customer of $10 million in one year and $4 million in five years. The yield curve is flat at 10%.
Required:
a. If it wants to fully fund and immunize its obligation to this customer with a single issue of a zero-coupon bond, what maturity bond must it purchase?
Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
b. What must be the face value and market value of that zero-coupon bond?
Note: Do not round intermediate calculations. Enter your answers in millions rounded to 2 decimal places.
$
$
a. Maturity of zero coupon bond
1.86
years
b. Face value
13.82
million
b. Market value
11.57
million
Explanation:
a. (1) Time until
Payment (Years)
(2) Cash Flow
(3) PV of CF (Discount Rate
= 10%)
(4) Weight
(5) Column (1) ×
Column (4)
1
$ 10
million
$ 9.09
million
0.7854
0.7854
5
4
million
2.48
million
0.2146
1.0729
Column Sums
$ 11.57
million
1.0000
1.8583
D
= 1.86 years = required maturity of zero-coupon bond.
b. The market value of the zero must be $11.57 million, the same as the market value of the obligations. Therefore, the face value must be:
$11.57 million × (1.10)
1.8583
= $13.82 million
Worksheet
Difficulty: 2 Intermediate
Source: Investments (Bodie, 13e, ISBN 1266836322) > Chapter 16: Managing Bond Portfolios > Chapter 16 Problems - Algorithmic & Static
References
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purchase?
Note: Do not round intermediate calculations. Round your answer
to 2 decimal places.
b. What must be the face value and market value of that zero-coupon
bond?
Note: Do not round intermediate calculations. Enter your answers in
millions rounded to 2 decimal places.
a. Maturity of zero coupon bond
b. Face value
b. Market value
years
million
million
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You must use formulas and cell references PV of Payments when computing all values in the green cells
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a)
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b)
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Show Transcribed Text
B) How could you construct a 1-year forward loan beginning in year 3? (Face Value)
C) How could you construct a 1-year forward loan beginning in year 4? (Face Value)
Required A Required B
Complete this question by entering your answers in the tabs below.
Face value
Rate of synthetic loan
→ Show Transcribed Text
Price
$ 970.93
898.39
836.92
How could you construct a 1-year forward loan beginning in year 3?
Note: Round your Rate of synthetic loan answer to 2 decimal places.
Required A
776.20
685.42
Required B
Face value
Rate of synthetic loan
Required C
7.85 %
Required C
How could you construct a 1-year forward loan beginning in year 4?
Note: Round your Rate of synthetic loan answer to 2 decimal places.
Ċ
13.29 %
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